Transmission strings

Harmonica generates light curves for transiting objects, such as exoplanets, where the sky-projected shape of these objects may deviate from circles. A given shape is defined by a single-valued function of angle around the objects terminator, called a transmission string. In the diagram below we illustrate a transmission string, \(r_{\rm{p}}(\theta)\), where \(r_{\rm{p}}\) is the effective radius of the object and \(\theta\) is the angle around the terminator from the object’s orbital velocity vector.

In Harmonica, a transmission string is parametrised in terms of a Fourier series. Mathematically we can write

\[r_{\rm{p}}(\theta) = \sum_{n=0}^{N_c} a_{n} \cos{(n \theta)} + \sum_{n=1}^{N_c} b_{n} \sin{(n \theta)},\]

where \(a_n\) and \(b_n\) are each \(n\rm{th}\) harmonic’s amplitude. The total number of terms is equal to \(2N_c + 1\). Below we show the shape contributions from the first 7 terms.

The above shapes demonstrate the basis (shown up to \(n=3\)) for generating various shapes. A transmission string may then be constructed from a linear combination of each shape contribution. To construct shapes of increasing complexity, more and more harmonics must be included. For further reference see Grant and Wakeford 2023.